Table of Contents
Closure algebras
Abbreviation: CloA
Definition
A \emph{closure algebra} is a modal algebra $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\diamond\rangle$ such that
$\diamond$ is \emph{closure operator}: $x\le \diamond x$, $\diamond\diamond x=\diamond x$
Remark: Closure algebras provide algebraic models for the modal logic S4. The operator $\diamond$ is the \emph{possibility operator}, and the \emph{necessity operator} $\Box$ is defined as $\Box x=\neg\diamond\neg x$.
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be closure algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:
$h(\diamond x)=\diamond h(x)$
Examples
Example 1: $\langle P(X),\cup,\emptyset,\cap,X,-,cl\rangle$, where $X$ is any topological space and $cl$ is the closure operator associated with $X$.
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Locally finite | no |
| Residual size | unbounded |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | yes, $n=2$ |
| Congruence regular | yes |
| Congruence uniform | yes |
| Congruence extension property | yes |
| Definable principal congruences | yes |
| Equationally def. pr. cong. | yes |
| Discriminator variety | no |
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
Finite members
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f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
